2. (35 points) You are designing a steel spherical tank (known as a 'Horton sphere') that...

 

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2. (35 points) You are designing a steel spherical tank (known as a 'Horton sphere') that will be used to store high-pressure H2 gas at 150.0 bars and 25.0C. The tank has an inner diameter of 8.0 m and a wall thickness of 355 mm. The concentration of H2 in the wall at the inner surface of the wall (r: T) is CA,1 = 1500 mol/m and negligible at the outer surface of the tank (r = 12). The diffusion = coefficient of H in the steel is DAB = 0.300 x 10-12 m/s. We can assume the hydrogen gas is an ideal gas. Assume diffusion is one-dimensional through the thickness of the tank wall. Analyze the system assuming steady state with constant pressure and temperature in the tank. There is no generation or consumption of H2 in the tank wall. Assume the total molar concentration C is constant and H2 is dilute in the tank wall. H gas T1 T2 (a) Starting from the governing Mass Diffusion Equation and using Fick's Law, find the simplified MDE in terms of the hydrogen molar concentration (CA) in the tank wall. (b) Solve the simplified equation in (a) for the general solution of the concentration profile C(r). (c) What are the appropriate boundary and/or initial conditions to solve for CA (r) in the tank wall. (d) Solve for the symbolic expression of the hydrogen concentration in the membrane, CA (r). (e) What is the molar flux of hydrogen at the exterior surface of the tank (r = r)? (f) What is the molar rate of hydrogen transfer at the exterior (r = r) and interior surface of the tank (r = r)? (g) If we now assume that the pressure in the tank will drop as hydrogen leaks out, what is the initial rate of pressure drop (dP/dt) within the tank resulting from hydrogen loss? 2. (35 points) You are designing a steel spherical tank (known as a 'Horton sphere') that will be used to store high-pressure H2 gas at 150.0 bars and 25.0C. The tank has an inner diameter of 8.0 m and a wall thickness of 355 mm. The concentration of H2 in the wall at the inner surface of the wall (r: T) is CA,1 = 1500 mol/m and negligible at the outer surface of the tank (r = 12). The diffusion = coefficient of H in the steel is DAB = 0.300 x 10-12 m/s. We can assume the hydrogen gas is an ideal gas. Assume diffusion is one-dimensional through the thickness of the tank wall. Analyze the system assuming steady state with constant pressure and temperature in the tank. There is no generation or consumption of H2 in the tank wall. Assume the total molar concentration C is constant and H2 is dilute in the tank wall. H gas T1 T2 (a) Starting from the governing Mass Diffusion Equation and using Fick's Law, find the simplified MDE in terms of the hydrogen molar concentration (CA) in the tank wall. (b) Solve the simplified equation in (a) for the general solution of the concentration profile C(r). (c) What are the appropriate boundary and/or initial conditions to solve for CA (r) in the tank wall. (d) Solve for the symbolic expression of the hydrogen concentration in the membrane, CA (r). (e) What is the molar flux of hydrogen at the exterior surface of the tank (r = r)? (f) What is the molar rate of hydrogen transfer at the exterior (r = r) and interior surface of the tank (r = r)? (g) If we now assume that the pressure in the tank will drop as hydrogen leaks out, what is the initial rate of pressure drop (dP/dt) within the tank resulting from hydrogen loss?